Comparison of different synchronousmachines for sensorless drives

Luigi AlbertiFaculty of Science and Technology

Free University of Bozen, Italyluigi.alberti@unibz.it

Nicola Bianchi, Silverio BolognaniDept. of Industrial Engineering

University of Padova, Italyedlab@dii.unipd.it

Abstract—The variation of the self-sensing capability of dif-ferent type of electrical machines due to parasitic effects isconsidered in this papers. Thanks to a specific simulation strategythe variations of the machine saliency and the error angle arecomputed considering different rotor positions and differentworking points. It is shown that a large variation occurs inparticular working points. The presented approach is usefulduring the design process of the machine in order to assessthe self-sensing capability of the machine and its impact on thecontrol scheme adopted in the drive.

I. INTRODUCTION

Sensorless techniques for the detection of the rotor positionhave been proposed some years ago in order to removethe position sensor in the electrical drive. Such a result isimportant for various reasons, among the others the reductionof the drive cost and the increase of the fault tolerant capabilityof the system.

The performance of the sensorless control depends on theelectrical machine geometry. In other words, a proper electricalmachine is necessary in order to guarantee an adequate self-sensing capability in the desired operating region.

Various models have been proposed in the literature toinvestigate the machine adopted in the sensorless drive, takinginto account also parasitic phenomena like the slot effectsand the machine non linearities [1]–[4]. It has also beenhighlighted that satisfactory performance is achieved by meansof a right combination of both sensorless control algorithmsand electrical machine geometry [5], [6].

This paper investigates the variation of the self-sensingcapability of different type of electrical machines due toparasitic effects. Adopting the simulation strategy proposed in[7], the variations of the machine saliency and the error angleare computed for different rotor positions and consideringdifferent working points. The presented approach is usefulduring the design process of the machine in order to assessthe self-sensing capability of the machine and its impact onthe control scheme adopted in the drive.

II. d-q MODELING OF ELECTRICAL MACHINE AT HIGHFREQUENCY

As far as the magnetic model of the electrical machine isconcerned, the classical d-q model in the rotor reference frameis used. The stator currents are assumed as state variables. Thed- and the q-axis flux linkages depend on both currents due to

cross saturation, i.e. they can be written as λsd(isd, isq) andλsq(isd, isq). Since the considered control technique is typi-cally applied at zero or quasi-zero speed, the same operationis considered here and the rotor speed ωe

m is assumed equalto zero.

A. Small-signal model

Let the working point of the motor be defined by the steady-state currents (Isd0, Isq0), i.e. by the selected state-variables.Then, δisd and δisq are small variations of the d- and q-axiscurrents around such a working point, so that the total currentresults as:

isd = Isd0 + δisd

isq = Isq0 + δisq(1)

Similarly, the flux linkages are expressed as:

λsd(isd, isq) = Λsd0 + δλsd

λsq(isd, isq) = Λsq0 + δλsq(2)

where Λsd0 and Λsq0 are the steady state flux linkages in theconsidered working point and the small variations are givenby:

δλsd = ldd δisd + ldq δisq

δλsq = lqd δisd + lqq δisq(3)

where ldd, lqq , ldq and lqd are the differential inductancesdefined as the partial derivative of the flux linkages evaluatedin the considered working point. Therefore, they are a functionof the working point (Isd0, Isq0). Thanks to the reciprocityproperty, and neglecting hysteretic phenomena, the differentialinductances ldq and lqd between the two axes are equal [8].

Taking into account both inductive and resistive part of thecurrents, the voltage equations can be written as [6], [7]:

vsd = R ısd + zdd ısd + zdq ısq

vsq = R ısq + zqd ısd + zqq ısq(4)

Considering the model described in (4), both the magnetic andresistive saliency of the electrical machine can be evaluated.Moreover, a better estimation of the error on the rotor angle isobtained since both the cross saturation and the eddy currentsare considered. Finally, the Joule losses associated to the eddycurrents due to the injected field are directly evaluated [9].

It is worth noticing that the voltages and currents in (4) arerepresent by phasors labeled with an overlined symbol. It is

978-1-4799-0224-8/13/$31.00 ©2013 IEEE 8220

ℜe

ℑm

ısd

ısq

vsd

vsq

(a) phasor diagram with injected ro-tating voltage and ldd < lqq , ldq =0

d

q

(b) current space vector trajectory inthe d-q plane

Fig. 1. high frequency current response due to rotating voltage injection

assumed that at steady-state, the high frequency voltages andcurrents are sinusoid.

III. HIGH FREQUENCY RESPONSE OF SYNCHRONOUSMACHINES

Equation (4) describes the machine high frequency responseat the injection frequency. For the computation of the machineself-sensing capability, it is particularly interesting the case ofrotating voltage injection. In fact, in such a case it is possibleto evaluate the machine response in a straightforward way asdescribed in the following.

A. Response of an ideal machine

Let us consider an ideal machine with Rs = 0, zdd =jωhf ldd zqq = jωhf lqq and zdq = zqd = 0. Then, (4) canbe rewritten as:

vsd = jωhf ldd ısd

vsq = jωhf lqq ısq(5)

Fig. 1(a) shows the phasor diagram described by (5): thetwo axes are independent and the currents lag the voltages of90 degrees. The d- and q-axis current amplitudes are differentdue to the different value of ldd and lqq. In the consideredexample it is ldd < lqq. The corresponding stator space vectordescribes the ellipse shown in Fig. 1(b).

B. Response with cross saturation

When cross saturation between the d- and q-axis is present,the voltage equation (4) can be written as:

vsd = jωhf ldd ısd + jωhf ldq ısq

vsq = jωhf lqd ısd + jωhf lqq ısq(6)

Fig. 2(a) shows the corresponding phasor diagram. In thiscase the currents are no longer orthogonal and the leadingangle of the voltages is not 90 degrees as in the previouscase. This imply that the ellipse described by the stator spacevector is changed in shape and orientation, as shows Fig. 2(b).

ℜe

ℑm

ısd

ısq

vsd

vsq

(a) phasor diagram with injected ro-tating voltage and ldd < lqq , ldq 6=0

d

q

(b) current space vector trajectory inthe d-q plane

Fig. 2. high frequency current response due to rotating voltage injection

C. General response

The high frequency response of the machine is in generaldescribed by (4). Also in this case the high frequency voltagesand currents can be represented by phasors as in the examplesabove. When the d- and q-axis currents are not orthogonal,symmetrical components can be adopted. This technique al-lows to split the general unbalanced current response as thesum of two balanced current systems. Such a technique hasbeen extensively adopted in the past, for example in the studyof the single phase induction motors [10].

Applying the symmetrical components theory, both d- andq-axis currents can be written as the sum of two components:

ısd = ıfsd + ıbsd

ısq = ıfsq + ıbsq(7)

ıfsd and ıfsq are the forward components, ıbsd and ıbsq are thebackward components. Such components are defined as:

ıfsd =ısd + jısq

2ıbsd =

ısd − jısq2

ıfsq =ısq − jısd

2ıbsq =

ısq + jısd2

(8)

As an example Fig. 3 shows the symmetrical componentsof the currents of Fig. 2(a). The forward components of thebalanced systems are orthogonal. The same for the backwardsystem. In the forward system, the d-axis current ıfsd leads theq-axis current ıfsq (as the applied voltage). In the backwardsystem the d-axis current ıbsd lags the q-axis current ıbsq .

Fig. 4 shows the trajectory in the d-q plane of the corre-sponding current space vector. Both the forward and backwardsymmetrical systems produce a rotating space vector in the d-q plane. The amplitude and the direction of rotation of the tworotating space vectors are different. The space vector of theforward system is labeled as −→ı f

d and it describes the biggercircle rotating in counter-clockwise direction. Similarly, thespace vector of the backward system is labeled −→ı f

b and itdescribes the smaller circle rotating in clockwise direction.The complete current response, which is an ellipse, is obtainedby the sum of the two space vectors, i.e. by the composition

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Fig. 3. Phasor diagram with symmetrical components

Fig. 4. high frequency current response due to rotating voltage injection

of the two circles as shown in Fig. 4 for three different timeinstant.

D. Computation of the machine saliency

Referring to the high frequency rotating voltage signalinjection, the saliency of the machine is defined as the ratiobetween the major and minor axis of the ellipse. From Fig.4it can be clearly computed as:

ξ =|−→ı f

d |+ |−→ıfb |

|−→ı fd | − |−→ı

fb |

=|ıfsd|+ |ıbsd||ıfsd| − |ıbsd|

(9)

The last relation is particularly useful to compute the saliencyduring the machine design and analysis. In the case ofFig. 1(b), i.e. when only ldd and lqq can be used to describethe machine saliency, (9) results in the well known ratioξ = lqq/ldd.

E. Computation of the error in the position estimation

The algorithms for the rotor position estimation are based onthe recognition of the machine saliency, i.e. on the tracking ofthe axis of the ellipse described by the high frequency currentspace vector. Due to parasitic effects, for example the crosssaturation between the d- and q-axis, such ellipse can rotaterespect to the d-axis of the angle ε so that an error in theestimation of the rotor position occurs. The error angle ε is

indicated in Fig. 4 and it can be readily computed from thesymmetrical systems as:

ε =arg(ıfsd)− arg(ıbsd)

2(10)

Also (10) can be easily implemented to compute the errorangle ε during the machine design and analysis.

The ellipse rotation is strictly related to the phase of thed- and q-axis phasors ısd and ısd. When they are orthogonal,as in Fig. 1(a), there is no rotation of the ellipse, i.e. ε = 0.When ısd leads ısq more than 90 degrees, the error angle isnegative, i.e. ε < 0. When ısd leads ısq less than 90 degrees,the error angle is positive, i.e. ε > 0.

The approach adopted here to compute the machine saliencyξ and the error angle ε is based on considering the relationsamong complex numbers, i.e. (9) and (10). The same result canbe achieved adopting different approaches, for example spacevectors. (9) and (10) are here preferred because assuming (4),i.e. sinusoidal quantities, the computation of the saliency andthe error angle is straightforward: the voltage are imposed atthe machine terminals in the FE simulations and the currentresponse is computed.

IV. EFFECT OF THE ROTOR POSITION

The parameters in (4) describe the voltage equation at themachine terminals. They are related to the field configurationinside the machine. For example they are affected by the satu-ration of iron or due to the slotting. Since these phenomena arerelated to the rotor position, a variation in the parameters areexpected varying the rotor angle. For example the harmonicsof the winding magneto motive force rotate asynchronouslyrespect to the rotor, producing variable local saturation in thevarious machine parts [11].

(a) ϑm = 0 deg (b) ϑm = 7.5 deg

Fig. 5. Flux density distribution for different rotor position in the inset motor.Operation at the nominal current.

As an example, Fig. 5 shows the flux density distributionunder load for two different rotor positions. The consideredmachine is an inset SPM motor which will be described inthe following Section. As can be noted, the saturation of theiron is quite different.

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Adopting the approach described above, i.e. using (9) and(10), the variation of the self-sensing capability of the machinewith the rotor position can be computed. The machine ismodeled using the procedure presented in [7]. The voltageequation (4) is directly included in the FE simulation usingcircuit-coupled equations.

A rotating voltage is imposed in the time-harmonic smallvariation simulation and the current is computed from the FEsimulation. Repeating the analysis for different rotor positionthe machine self-sensing capability is computed taking intoaccount the effects of secondary effects like machine slotting,winding magneto motive force harmonics and so on. All theseaspects are sometimes referred to as “secondary saliency”.

V. APPLICATION EXAMPLES

In the this Section the strategy presented above is appliedto compute the self-sensing capability of different types ofmachine. In particular both the saliency, computed using (9),and the error angle, computed using (10), are calculated fordifferent rotor positions.

A. Inset motor

At first, the inset SPM motor is considered. Fig. 6 shows asketch of the machine geometry. It is a 12-slot 8-pole machineso that it is characterized by a fractional-slot non-overlappedwinding [12], [13]. In particular the number of slots per pole isequal to 3/2. The inset motor is derived from an SPM machinewith additional iron teeth in the rotor between each couple ofmagnets. Such teeth introduce a detectable anisotropy which isrobust with the operating conditions. Nevertheless, due to thenonlinear characteristic of the iron, a cross saturation effectis expected, which yield, in turn, an error angle ε as reportedin [14]. Additional information about such a machine and itsdesign can be found in [15].

Fig. 6. The geometry of the fractional-slot inset motor

Fig. 7 shows the computed saliency and error angle as afunction of the rotor position considering various currents.In particular operations at no-load, at the nominal current

0.8

1.0

1.2

1.4

1.6

1.8

0 3 6 9 12 15

ξ

rotor position (mech deg)

0

In

2In

(a) Variation of the machine saliency with the rotor position

-6-30369

121518

0 3 6 9 12 15ε(deg)

rotor position (mech deg)

In

2In

0

(b) Variation of the error angle with the rotor position

Fig. 7. Simulation results for the inset motor

and at twice the nominal current have been considered. Asexpected such characteristics are periodic with a period of15 mechanical degrees (i.e. 60 electrical degrees).

Fig. 7(a) shows the saliency ξ: it is always greater than onewhich is an important condition to ensure a proper trackingof the rotor position. There is a saliency oscillation with therotor position.

Fig. 7(b) shows the error angle ε. At no load, the error angleoscillates around zero. Increasing the current, the average errorangle increases along with the oscillation amplitude. The errorangle is considerable, especially at twice the nominal current.

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

-3.0 -2.0 -1.0 0.0 1.0 2.0 3.0

d

qε=-3.4 degξ=1.520

ϑm=3.000

(a) no-load

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

-3.0 -2.0 -1.0 0.0 1.0 2.0 3.0

d

qε=16.1 degξ=1.233

ϑm=8.250

(b) 2In

Fig. 8. Current trajectory for the inset motor in two different working point

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Fig. 8 shows the high frequency current trajectory in the d-q plane in two different working points. The origin of the planeis refereed to the actual working point. Fig. 8(a) shows theellipse at no-load operation for the rotor angle ϑm = 4 deg. InFig. 8(b) operations at twice the nominal current is considered.Such ellipse refers to the rotor angle ϑm = 11 deg. Thedifference in the shape and in the orientation of the two ellipsesis evident.

B. Interior permanent magnet motor

An IPM machine with fractional-slot concentrated windingis considered hereafter. Fig. 9 shows a sketch of the consideredmachine. The machine design is detailed in [16]. Thanks toits rotor anisotropy, good self-sensing capability are expected.Nevertheless this have to be carefully checked in the operatingregion, as described in the following.

Fig. 9. The geometry of the fractional-slot “ISA” IPM motor

Fig. 10 shows the computed saliency and error angle as afunction of the rotor position considering various currents.

Fig. 10(a) shows the saliency ξ versus the rotor position. Inthis case, there are some rotor positions at the nominal currentwhere the saliency becomes equal to one. The machine loosesits self sensing capability. In such cases, the correspondingerror angle exhibits a discontinuity as shown in Fig. 10(b).This is a critical working condition for the sensorless drive.

Fig. 11 shows the high frequency current trajectory in thed-q plane in two different working points. Fig. 11(a) showsthe ellipse in the critical operating point at nominal current.The ellipse degenerates in a circle and so the self-sensingcapability of the machine is lost. In Fig. 11(b), operationsat twice the nominal current for the rotor angle ϑm = 4.5 degis considered. In this case the machine exhibits an adequatesaliency and a considerable error angle.

C. Interior permanent magnet “Machaon” motor

At last an IPM machine with integral-slot winding is con-sidered. It is characterized by unsymmetrical rotor barriers

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 5 10 15 20 25 30

ξ

rotor position (mech deg)

0

In

2In

(a) Variation of the machine saliency with the rotor position

-120-100-80-60-40-200

204060

0 5 10 15 20 25 30ε(deg)

rotor position (mech deg)

0In

2In

(b) Variation of the error angle with the rotor position

Fig. 10. Simulation results for the fractional-slot “ISA” IPM motor

-2.00

-1.00

0.00

1.00

2.00

-2.00 -1.00 0.00 1.00 2.00

d

q

ε=-110.6 degξ=1.026

ϑm=14.5

(a) In

-2.00

-1.00

0.00

1.00

2.00

-2.00 -1.00 0.00 1.00 2.00

d

q

ε=-58.1 degξ=2.407

ϑm=4.5

(b) 2In

Fig. 11. Current trajectory for the IPM ISA motor in two different workingpoint

in order to reduce the torque ripple and it is referred to as“Machaon” rotor [17]. Fig. 12 shows the considered geometry.

Fig. 13 shows the computed saliency and error angle as afunction of the rotor position considering different currents.Fig. 13(a) shows the saliency ξ. In this case the periodicity isno longer exactly 60 electrical degrees (i.e. 30 mechanicaldegrees) due to the unsymmetrical rotor barriers. For thedifferent working conditions considered, the machine saliencyremains always considerable, so that the machine does notloose its self-sensing capability.

Fig. 13(b) shows the computed error angle. Its oscillationsincrease with the current, but the error angle is always limitedcompared to the other machines.

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Fig. 12. The geometry of the “Machaon” IPM motor

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

0 10 20 30 40 50 60

ξ

rotor position (mech deg)

0

In

2In

(a) Variation of the machine saliency with the rotor position

-5

-4

-3

-2

-1

0

1

0 10 20 30 40 50 60

ε(deg)

rotor position (mech deg)

0

In

2In

(b) Variation of the error angle with the rotor position

Fig. 13. Simulation results for the “Machaon” IPM motor

Fig. 14 shows the high frequency current trajectory in thed-q plane in two different working points. Fig. 14(a) showsthe ellipse at no-load for the rotor angle ϑm = 4.0 deg. InFig. 11(b) operations at twice the nominal current for the rotorangle ϑm = 10 deg is considered. In both cases it is visiblethat a high saliency is exhibited by the motor with a limitedrotation of the ellipse.

VI. CONCLUSION

In this paper the self-sensing capability of different type ofmachines have been computed and compared. The effects ofthe rotor position and the working point have been computed.

-0.10

-0.05

0.00

0.05

0.10

-0.10 -0.05 0.00 0.05 0.10

d

q

ε=0.5 degξ=6.168

ϑm=4.000

(a) no load

-0.10

-0.05

0.00

0.05

0.10

-0.10-0.05 0.00 0.05 0.10

d

q

ε=-3.4 degξ=2.996

ϑm=10.000

(b) 2In

Fig. 14. Current trajectory for the IPM “Machaon” motor in two differentworking point

It has been shown that a large variation exists, in particulardue to saturation of the iron. In same cases, for specific rotorposition the self-sensing capability of the machine is lost,being the saliency equal to zero

The approach adopted in this paper is useful during thedesign process of the machine in order to assess the self-sensing capability of the machine and its impact on the controlscheme adopted in the drive.

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